A071904 -id:A071904 - cp's OEIS frontend

A269345 Smaller of two consecutive odd numbers that are composites.

25, 33, 49, 55, 63, 75, 85, 91, 93, 115, 117, 119, 121, 123, 133, 141, 143, 145, 153, 159, 169, 175, 183, 185, 187, 201, 203, 205, 207, 213, 215, 217, 219, 235, 243, 245, 247, 253, 259, 265, 273, 285, 287, 289, 295, 297, 299, 301, 303, 319, 321, 323, 325, 327, 333

Offset: 1

Waldemar Puszkarz, Feb 24 2016

nonn

Analogous to A001359 for odd composite numbers (A071904).
Consists of numbers that cannot be the difference of two primes: an odd number m can be the difference of two primes only if m+2 is prime, which cannot be the case for any a(n) as a(n)+2 is composite.
Some terms form subsequences of perfect powers, e.g., A106564 (for squares) and A269346 (for cubes).
Any composite of the form 6k+1 (A016921) is a term: (6k+1)+2 = 3(2k+1) is both odd and composite as a product of two odd numbers, thus 6k+1, being odd, is a term if it is composite.

			25 belongs to this sequence because 27=25+2 is the next odd composite.

Eric Weisstein's World of Mathematics, Twin Composites

Cf. A071904 (odd composites), A001359 (similar sequence for primes).
Cf. A106564, A269346.
Cf. A061673.

[n: n in [1..350]| not IsPrime(n) and not IsPrime(n+2) and n mod 2 eq 1]; // Vincenzo Librandi, Feb 28 2016

Select[Range[450], OddQ[#]&& !PrimeQ[#]&&!PrimeQ[#+2]&]

for(n=1, 450, n%2==1&&!isprime(n)&&!isprime(n+2)&&print1(n, ", "))

a(n) = A061673(n) - 1. - M. F. Hasler, Nov 18 2018

Name edited by Michel Marcus, Jul 27 2023

A339517 Odd composite integers m such that A000032(2*m-J(m,5)) == J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

Original entry on oeis.org

323, 377, 1001, 1183, 1729, 1891, 3827, 4181, 5777, 6601, 6721, 8149, 8841, 10877, 11663, 13201, 13981, 15251, 17119, 17711, 18407, 19043, 23407, 25877, 26011, 27323, 30889, 34561, 34943, 35207, 39203, 40501, 41041

Offset: 1

Ovidiu Bagdasar, Dec 07 2020

nonn

The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy V(k*p-J(p,D)) == V(k-1)*J(p,D) (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4.
The composite integers m with the property V(k*m-J(m,D)) == V(k-1)*J(m,D) (mod m) are called generalized Pell-Lucas pseudoprimes of level k- and parameter a.
Here b=-1, a=1, D=5 and k=2, while V(m) recovers A000032(m) (Lucas numbers).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica and Ovidiu Bagdasar, On Generalized Lucas Pseudoprimality of Level k, Mathematics (2021) Vol. 9, 838.
Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 24(1), 9-15 (2018).

Cf. A000032, A071904, A339125 (a=1, b=-1, k=1).

Cf. A339518 (a=3, b=-1), A339519 (a=5, b=-1), A339520 (a=7, b=-1).

Select[Range[3, 45000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[LucasL[2*# - JacobiSymbol[#, 5]] - JacobiSymbol[#, 5], #] &]

A339518 Odd composite integers m such that A006497(2m-J(m,13)) == 3J(m,13) (mod m), where J(m,13) is the Jacobi symbol.

Original entry on oeis.org

15, 75, 105, 119, 165, 255, 375, 649, 1189, 1635, 1763, 1785, 1875, 2233, 2625, 3599, 3815, 4125, 4187, 5475, 5559, 5887, 6375, 6601, 6681, 7905, 8175, 9265, 9375, 9471, 11175, 11767, 11977, 12095, 12403, 12685, 12871, 13601, 14041, 14279, 15051, 16109, 16359

Offset: 1

Ovidiu Bagdasar, Dec 07 2020

nonn

The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy V(k*p-J(p,D)) == V(k-1)*J(p,D) (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4.
The composite integers m with the property V(k*m-J(m,D)) == V(k-1)*J(m,D) (mod m) are called generalized Pell-Lucas pseudoprimes of level k- and parameter a.
Here b=-1, a=3, D=13 and k=2, while V(m) recovers A006497(m).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 24(1), 9-15 (2018).

Cf. A006497, A071904, A339126 (a=3, b=-1, k=1).

Cf. A339517 (a=1, b=-1), A339519 (a=5, b=-1), A339520 (a=7, b=-1).

Select[Range[3, 20000, 2], CoprimeQ[#, 13] && CompositeQ[#] && Divisible[LucasL[2*# - JacobiSymbol[#, 13], 3] - 3*JacobiSymbol[#, 13], #] &]

A339520 Odd composite integers m such that A086902(2m-J(m,53)) == 7J(m,53) (mod m), where J(m,53) is the Jacobi symbol.

Original entry on oeis.org

25, 35, 51, 65, 75, 91, 105, 175, 203, 325, 391, 455, 575, 645, 861, 1247, 1275, 1295, 1633, 1763, 1775, 1785, 1875, 1921, 2275, 2407, 2415, 2599, 2625, 2651, 3045, 3367, 4199, 4579, 4623, 5629, 5835, 5887, 6441, 6699, 9959, 10465, 10815, 10825, 10877, 11865, 12025

Offset: 1

Ovidiu Bagdasar, Dec 07 2020

nonn

The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy V(k*p-J(p,D)) == V(k-1)*J(p,D) (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4.
The composite integers m with the property V(k*m-J(m,D)) == V(k-1)*J(m,D) (mod m) are called generalized Pell-Lucas pseudoprimes of level k- and parameter a.
Here b=-1, a=7, D=53 and k=2, while V(m) recovers A086902(m).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 24(1), 9-15 (2018).

Cf. A086902, A071904, A339128 (a=7, b=-1, k=1).

Cf. A339517 (a=1, b=-1), A339518 (a=3, b=-1), A339529 (a=5, b=-1).

Select[Range[3, 20000, 2], CoprimeQ[#, 53] && CompositeQ[#] && Divisible[LucasL[2*# - JacobiSymbol[#, 53], 7] - 7*JacobiSymbol[#, 53], #] &]

A339522 Odd composite integers m such that A003501(2*m-J(m,21)) == 5 (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.

Original entry on oeis.org

95, 115, 145, 253, 391, 527, 551, 713, 715, 779, 935, 1045, 1615, 1705, 1805, 1807, 1919, 2185, 2627, 2755, 2893, 2929, 2945, 3281, 4033, 4141, 4205, 5191, 5671, 5777, 5983, 6049, 6479, 7645, 7739, 8441, 8555, 8695, 9361, 11663, 11815, 12121, 12209, 12265, 14491

Offset: 1

Ovidiu Bagdasar, Dec 07 2020

nonn

The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy V(k*p-J(p,D)) == V(k-1) (mod p) whenever p is prime, k is a positive integer, b=1 and D=a^2-4.
The composite integers m with the property V(k*m-J(m,D)) == V(k-1) (mod m) are called generalized Pell-Lucas pseudoprimes of level k+ and parameter a.
Here b=1, a=5, D=21 and k=2, while V(m) recovers A003501(m).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Robert Israel, Table of n, a(n) for n = 1..1000
Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 24(1), 9-15 (2018).

Cf. A003501, A071904, A339130 (a=5, b=1, k=1).

Cf. A339521 (a=3, b=1), A339523 (a=7, b=1).

filter:= proc(m)
uses LinearAlgebra:-Modular;
local p,M;
  if igcd(m,21) <> 1 then return false fi;
  if isprime(m) then return false fi;
  p:= 2*m - numtheory:-jacobi(m,21);
  M:= Mod(m,[[0,1],[-1,5]],integer[8]);
  (MatrixPower(m,M,p) . <2,5>)[1] - 5 mod m = 0
end proc:
select(filter, [seq(i,i=9..20000,2)]); # Robert Israel, Dec 15 2020

Select[Range[3, 20000, 2], CoprimeQ[#, 21] && CompositeQ[#] && Divisible[2*ChebyshevT[2*# - JacobiSymbol[#, 21], 5/2] - 5, #] &]

A339523 Odd composite integers m such that A056854(2*m-J(m,45)) == 7 (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.

Original entry on oeis.org

91, 203, 323, 329, 377, 451, 1001, 1081, 1183, 1547, 1729, 1771, 1819, 1891, 1967, 2033, 2093, 2639, 2821, 3197, 3311, 3653, 3731, 3827, 4181, 4669, 5551, 5671, 5777, 5887, 6601, 6721, 7471, 7931, 7973, 8149, 8557, 9541, 9737, 10877, 11309, 11663, 11977, 13201

Offset: 1

Ovidiu Bagdasar, Dec 07 2020

nonn

The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy V(k*p-J(p,D)) == V(k-1) (mod p) whenever p is prime, k is a positive integer, b=1 and D=a^2-4.
The composite integers m with the property V(k*m-J(m,D)) == V(k-1) (mod m) are called generalized Pell-Lucas pseudoprimes of level k+ and parameter a.
Here b=1, a=7, D=45 and k=2, while V(m) recovers A056854(m).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Robert Israel, Table of n, a(n) for n = 1..1000
Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 24(1), 9-15 (2018).

Cf. A056854, A071904, A339131 (a=7, b=1, k=1).

Cf. A339521 (a=3, b=1), A339522 (a=5, b=1).

filter:= proc(m)
uses LinearAlgebra:-Modular;
local p,M;
  if igcd(m,45) <> 1 then return false fi;
  if isprime(m) then return false fi;
  p:= 2*m - numtheory:-jacobi(m,45);
  M:= Mod(m,[[0,1],[-1,7]],integer[8]);
  (MatrixPower(m,M,p) . <2,7>)[1] - 7 mod m = 0
end proc:
select(filter, [seq(i,i=9..10000,2)]); # Robert Israel, Dec 15 2020

Select[Range[3, 15000, 2], CoprimeQ[#, 45] && CompositeQ[#] && Divisible[LucasL[4*(2*# - JacobiSymbol[#, 45])] - 7, #] &]

A340118 Odd composite integers m such that A000045(2*m-J(m,5)) == 1 (mod m), where J(m,5) is the Jacobi symbol.

Original entry on oeis.org

323, 377, 609, 1891, 3081, 3827, 4181, 5777, 5887, 6601, 6721, 8149, 10877, 11663, 13201, 13601, 13981, 15251, 17119, 17711, 18407, 19043, 23407, 25877, 27323, 28441, 28623, 30889, 32509, 34561, 34943, 35207, 39203, 40501

Offset: 1

Ovidiu Bagdasar, Dec 28 2020

nonn

The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(2*p-J(p,D)) == 1 (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4.

The composite integers m with the property U(k*m-J(m,D)) == U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a. Here b=-1, a=1, D=5 and k=2, while U(m) is A000045(m) (Fibonacci sequence).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.

Cf. A000045, A071904, A081264 (a=1, b=-1, k=1), A327653 (a=3, b=-1, k=1).

Cf. A340119 (a=3, b=-1, k=2), A340120 (a=5, b=-1, k=2), A340121 (a=7, b=-1, k=2).

Select[Range[3, 50000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[Fibonacci[2*#-JacobiSymbol[#, 5], 1] - 1, #] &]

A340119 Odd composite integers m such that A006190(2*m-J(m,13)) == 1 (mod m), where J(m,13) is the Jacobi symbol.

Original entry on oeis.org

9, 27, 63, 81, 99, 119, 153, 243, 567, 649, 729, 759, 891, 903, 1071, 1189, 1377, 1431, 1539, 1763, 1881, 1953, 2133, 2187, 3599, 3897, 4187, 4585, 5103, 5313, 5559, 5589, 5819, 6561, 6681, 6831, 6993, 8019, 8127, 8829, 8855, 9639, 9999, 10611, 11135, 11691, 11961

Offset: 1

Ovidiu Bagdasar, Dec 28 2020

nonn

The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(2*p-J(p,D)) == 1 (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4. The composite integers m with the property U(k*m-J(m,D)) == U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a. Here b=-1, a=3, D=13 and k=2, while U(m) is A006190(m).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.

Cf. A006190, A071904, A081264 (a=1, b=-1, k=1), A327653 (a=3, b=-1, k=1).

Cf. A340118 (a=1, b=-1, k=2), A340120 (a=5, b=-1, k=2), A340121 (a=7, b=-1, k=2).

Select[Range[3, 12000, 2], CoprimeQ[#, 13] && CompositeQ[#] && Divisible[Fibonacci[2*#-JacobiSymbol[#, 13], 3] - 1, #] &]

A340120 Odd composite integers m such that A052918(2*m-J(m,29)) == 1 (mod m), where J(m,29) is the Jacobi symbol.

Original entry on oeis.org

9, 15, 25, 27, 45, 75, 91, 121, 125, 135, 143, 147, 175, 225, 275, 325, 375, 441, 483, 625, 675, 735, 755, 1125, 1323, 1547, 1573, 1875, 1935, 2015, 2205, 2275, 2485, 2943, 3025, 3125, 3375, 3575, 3675, 3775, 3843, 4375, 5525, 5625, 5819, 6543, 6615, 6721

Offset: 1

Ovidiu Bagdasar, Dec 28 2020

nonn

The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(2*p-J(p,D)) == 1 (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4.

The composite integers m with the property U(k*m-J(m,D)) == U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a. Here b=-1, a=5, D=29 and k=2, while U(m) is A052918(m).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.

Cf. A052918, A071904, A340095 (a=5, b=-1, k=1).

Cf. A340118 (a=1, b=-1, k=2), A340119 (a=3, b=-1, k=2), A340121 (a=7, b=-1, k=2).

Select[Range[3, 15000, 2], CoprimeQ[#, 29] && CompositeQ[#] && Divisible[Fibonacci[2*#-JacobiSymbol[#, 29], 5] - 1, #] &]

A340121 Odd composite integers m such that A054413(2*m-J(m,53)) == 1 (mod m), where J(m,53) is the Jacobi symbol.

Original entry on oeis.org

25, 35, 39, 49, 51, 65, 91, 147, 175, 245, 301, 325, 343, 391, 455, 507, 575, 605, 637, 663, 741, 833, 897, 903, 935, 1127, 1205, 1225, 1247, 1295, 1505, 1595, 1633, 1715, 1763, 1775, 1911, 1921, 2107, 2275, 2401, 2407, 2499, 2599, 2651, 3025, 3143, 3185, 3311

Offset: 1

Ovidiu Bagdasar, Dec 28 2020

nonn

The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy U(2*p-J(p,D)) == 1 (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4.

The composite integers m with the property U(k*m-J(m,D)) == U(k-1) (mod m) are called generalized Lucas pseudoprimes of level k- and parameter a. Here b=-1, a=7, D=53 and k=2, while U(m) is A054413(m).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.

Cf. A054413, A071904, A340096 (a=7, b=-1, k=1).

Cf. A340118 (a=1, b=-1, k=2), A340119 (a=3, b=-1, k=2), A340120 (a=5, b=-1, k=2).

Select[Range[3, 10000, 2], CoprimeQ[#, 53] && CompositeQ[#] && Divisible[Fibonacci[2*#-JacobiSymbol[#, 53], 7] - 1, #] &]

cp's OEIS Frontend

A269345 Smaller of two consecutive odd numbers that are composites.

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A339517 Odd composite integers m such that A000032(2*m-J(m,5)) == J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

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A339518 Odd composite integers m such that A006497(2*m-J(m,13)) == 3*J(m,13) (mod m), where J(m,13) is the Jacobi symbol.

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A339520 Odd composite integers m such that A086902(2*m-J(m,53)) == 7*J(m,53) (mod m), where J(m,53) is the Jacobi symbol.

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A339522 Odd composite integers m such that A003501(2*m-J(m,21)) == 5 (mod m) and gcd(m,21)=1, where J(m,21) is the Jacobi symbol.

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A339523 Odd composite integers m such that A056854(2*m-J(m,45)) == 7 (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.

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A340118 Odd composite integers m such that A000045(2*m-J(m,5)) == 1 (mod m), where J(m,5) is the Jacobi symbol.

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A340119 Odd composite integers m such that A006190(2*m-J(m,13)) == 1 (mod m), where J(m,13) is the Jacobi symbol.

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A339518 Odd composite integers m such that A006497(2m-J(m,13)) == 3J(m,13) (mod m), where J(m,13) is the Jacobi symbol.

A339520 Odd composite integers m such that A086902(2m-J(m,53)) == 7J(m,53) (mod m), where J(m,53) is the Jacobi symbol.